Linear method for the approximation of differentiable functions

This article is devoted to the problem of the approximation of functions in the metric of space L_p(0, 1), the s-th derivative of the functions being continuous and the (s + l)-th derivative belonging to the space L_q(0, 1), with p, q 1, by spline functions of order s with fixed almost uniform nodes.Translated from Matematicheskie Zametki, Vol. 7, No. 4, pp. 423–430, April, 1970.     

An interpolation method for the solution of identification problems for a multidimensional functional system described by a Urysohn operator

An algorithm is constructed for identifying deterministic dynamical systems described using a Urysohn operator, i. e., the investigation is conducted under the hypothesis that the investigated object has a partially known structure, a grey box, and the identification problem consists in approximating the Urysohn nonlinearity. The initial information for solving the identification problem is the aggregate of the input effects on a physical object, the input, and also its reaction to these effects, the output. The accuracy of the algorithm is discussed.Translated from Vychislitel'naya i prikladnaya matematika, No. 70, pp. 31–36, 1990.     

A new characterization of <Emphasis Type="Italic">PSU</Emphasis>(<Emphasis Type="Italic">p,q</Emphasis>)

Summary Let G be a finite group. Order components of G were introduced in Chen [5]. Let OC(G) be the set of order components of G. Some finite groups are characterizable by their order components. This assertion was proved for the simple groups PSU(p,q), where p=3, 5, 7 and 11. In this paper, we prove that the simple groups PSU(p,q) can be uniquely determined by their order components, where p&ge;13 is a prime number. Main consequences of our results are the validity of a conjecture of J. G. Thompson and another conjecture of W. Shi and J. Bi for the groups under consideration.     

Parabolic subgroups of Chevalley groups over a semilocal ring

Let G be the Chevalley group over a commutative semilocal ring R which is associated with a root system . The parabolic subgroups of G are described in the work. A system =() of ideals in R ( runs through all roots of the system ) is called a net of ideals in the commutative ring R if ₊ for all those roots and for which + is also a root. A net is called parabolic if =R for >0. The main theorem: under minor additional assumptions all parabolic subgroups of G are in bijective correspondence with all parabolic nets . The paper is related to two works of K. Suzuki in which the parabolic subgroups of G are described under more stringent conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 43–58, 1978.     

Differentiation of fractional order onp-groups,approximation properties

G- p- . [5] - (G) L _r(G) (1r<), . . , - . , , , . . , X. , . (. [1], [2] [4]).     

Approximations in thermal explosion theory and the nature of the degenerate critical point

Summary In studies of thermal explosion the Frank-Kamenetskii approximation sets exp(–E/RT)=exp(–E/RT ₀)exp (/(1+))exp(–E/RT ₀)exp, where=RT ₀/E i.e. it assumes0. When this approximation is not made, it is known that criticality vanishes for greater than a certain value _*, say. This may occur whether the Arrhenius form is used or some suitable approximation to it; many authors have proposed approximations involving the maximum dimensionless temperature in the reactant. The nature of the degeneracy near the value _* is examined for such approximations in general, some approximations are considered and the results compared.

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Zusammenfassung In Studien von thermischen Explosionen setzt man in der Näherung von Frank-Kamenetskii exp(–E/RT)=exp(–E/RT ₀)exp(/(1+))exp(–E/RT ₀) exp), wobei=RT ₀/E ist, d.h. man nimmt0 an. Wenn diese Näherung nicht benützt wird, so weiß man, daß die Kritikalität verschwindet wenn einen gewissen Wert _* überschreitet. Dies findet man mit Benützung der Formel von Arrhenius oder mit einer Näherung dazu; viele Autoren haben Näherungen vorgeschlagen mit Verwendung der maximalen dimensionslosen Temperatur im Reaktionsgemisch. Es wird für solche Näherungen die Natur der Entartung der Lösung in der Umgebung von _* untersucht; die Resultate für verschiedene Näherungen werden verglichen.

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Résumé Dans les études de la théorie de l'explosion thermale l'approximation de Frank-Kamenetskii pose exp(–E/RT)=exp(–E/RT ₀)exp(/(1+)]exp(–E/RT ₀) exp, avec=RT ₀/E, c'est à dire on admet0.On sait que, en dehors de cette approximation, la limite critique disparaît lorsque dépasse une certaine valeur dénommée _*. Ceci peut se produire soit en utilisant la forme d'Arrhenius ou une approximation adéquate.Plusieurs auteurs ont proposé des approximations utilisant la température maximum non-dimensionelle du réactif. Dans la présente étude on examine d'une façon générale le caractère de la dégénérescence aux alentours de la valeur _* pour ce genre d'approximations. Ensuite on considère quelques approximations particulières et les résultats sont comparés.

     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

Strong coupling limit for a certain class of polaron models

Summary It will be shown, that a Polaron-like asymptotic is given for a class of functions which are called of decreasing interaction and nicely approximable with respect to the Wiener process. Using Large Deviation techniques one can see, that the asymptotics for thestrong coupling limit for those Polaron models is given by a meanfield model and can be described by a variational problem.     

On strong summability of polynomial expansions

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Dedicated to Professor L. Leindler on his 50^th birthday     

1-formes fermées singulières et groupe fondamental

Summary We study the influence of the ₁ of a closed manifoldM ⁿ (n3) on the foliations ofM defined by closed differential 1-forms with Morse singularities (of index 0,n). Every nonexact form is cohomologous to a weakly complete one, that is one whose leaf space is of the same type as that of a nonsingular form. Generically, a form has compact leaves or is weakly complete. If ₁ M has no quotient isomorphic to *, then every nonexact form onM is weakly complete. We also say a form is complete if every path inM is homotopic to either a path transverse to or a path contained in a leaf of . Completeness of depends only on its de R ham cohomology class. The set of complete cohomology classes depends only on ₁ M and is related to finitely generated normal subgroups of ₁ M with quotient . If ₁ M is nilpotent (or even polycyclic), every nonexact form onM is complete. On irreducible 3-manifolds, a form is complete iff it is cohomologous to a nonsingular one.     

Algorithms for optimization problems with exclusion constraints

This paper proposes algorithms for minimizing a continuously differentiable functionf(x): ⁿ subject to the constraint thatx does not lie in specified bounded subsets of ⁿ . Such problems arise in a variety of applications, such as tolerance design of electronic circuits and obstacle avoidance in the selection of trajectories for robot arms. Such constraints have the form . The function is not continuously differentiable. Algorithms based on the use of generalized gradients have considerable disadvantages because of the local concavity of at points where the set {j|g ^j (x)=(x)} has more than one element. Algorithms which avoid these disadvantrages are presented, and their convergence is established.This research was sponsored in part by the National Science Foundation under Grant ECS-81-21149, the Air Force Office of Scientific Research (AFSC), United States Air Force under Contract F49620-79-C-0178, the Office of Naval Research under Grant N00014-83-K-0602, the Air Force Office of Scientific Research under Grant AFOSR-83-0361, and the Semiconductor Research Consortium under Grant SRC-82-11-008.     

О Перенумерации Графов

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Applications of matrix functions

A, ()=0, c(z)- T - TY=AY. A y(t)=y(, t) Ty=y Y(t). - c(T)y=0.     

On Fourier multipliers between Besov spaces with 0 <p o ≤ min{1,p 1}

, 0<0 min{1,p1}.     

The method of Boltzmann collision integral eigenfunctions in the kinetic theory of long-wave fluctuations

Linearized general equations of long-wave fluctuation kinetics are solved (utilizing eigenfunctions and eigenvalues of the linearized Boltzmann collision integral) in the asymptotic region t (_r is the relaxation time). A general form for linearized equations of the fluctuation hydrodynamics is obtained. Effective initial conditions for the fluctuation hydrodynamics equations are derived for the case where fluctuations of any order are absent at the initial moment. The time asymptotics of the one-particle distribution function are found at the evolutionary stage of the fluctuations where the fluctuations of hydrodynamic quantities play an essential role. This is compared with results of the long hydrodynamic tails theory obtained earlier.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 3, pp. 469–488, March, 1996.Translated by V. I. Serdobolskii.     

On Sets of Planes in Projective Spaces Intersecting Mutually in One Point

Let be a projective space. In this paper we consider sets of planes of such that any two planes of intersect in exactly one point. Our investigation will lead to a classification of these sets in most cases. There are the following two main results:- If is a set of planes of a projective space intersecting mutually in one point, then the set of intersection points spans a subspace of dimension 6. There are up to isomorphism only three sets where this dimension is 6. These sets are related to the Fano plane.- If is a set of planes of PG(d,q) intersecting mutually in one point, and if q3, 3(q²+q+1), then is either contained in a Klein quadric in PG(5,q), or is a dual partial spread in PG(4,q), or all elements of pass through a common point.     

Methoden der Punktebewertung für Kreditscoringsysteme

Zusammenfassung Kreditscoringsysteme für Konsumentenkredite sollen eine erste Einstufung des bei einem potentiellen Kunden zu erwartenden Kreditrisikos ermöglichen. Hierzu werden Kundenmerkmale numerisch quantifiziert und zu einem Bonitätsscore addiert. Anhand von konkretem Datenmaterial werden solche Quantifizierungsoder Punktebewertungsverfahren diskutiert, Verbindungen der Punkteaddition mit der Unabhängigkeitsregel zur statistischen Diskrimination hergestellt und für die praktische Eisetzbarkeit notwendige Modifikationen beschrieben. Es zeigt sich auch, daß die bisweilen vorgeschlagenen multivariaten Verfahren wie lineare oder quadratische Diskriminanzanalysen im Vergleich zur Punkteaddition keine für die Praxis relevanten Verbesserungen erbringen, wenn die Punktebewertung an diejenige der Unabhängigkeitsregel oder an die der sog. nullpunktzentrierten Punktezuordnung angepaßt wird, wobei letztere Methode sogar von mathematischen Laien (z.B. in der zuständigen Fachabteilung) leicht und intuitiv einsehbar ist.

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Credit scoring systems for consumer loans should be able to estimate approximately an applicant's creditworthiness. To this end one quantifies numerically the items (variables) and adds them up to a creditworthiness score. Based upon concrete data such methods of scaling items are discussed, connections of the score adding method with the independence rule of statistical discrimination are explained and it is shown how to modify the scaling methods for practical use. Another result is that multivariate methods as discriminant analysis don't improve the misclassification rates of the simple score adding rule relevantly when the scaling of the items is adapted to the scaling of the independence rule or to the so-called zero-centered scaling method; this latter method is intuitively understandable even for mathematical laymen.

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Ausführliche Fassung eines Vortrags bei der DGOR-Tagung 1980 in Essen.     

A Hubert space characterization among function spaces

— [0,1] ,E — - e=1 [0,1]. I — _E =1, E=L ₂ x _e =xL ₂ x E.

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This work was prepared when the second author was a visiting professor of the CNR at the University of Firenze. He was supported by the Soros International Fund.     

A new condensation principle

We generalize (A), which was introduced in [Sch], to larger cardinals. For a regular cardinal >₀ we denote by (A) the statement that and for all regular >,is stationary in It was shown in [Sch] that can hold in a set-generic extension of L. We here prove that can hold in a set-generic extension of L as well. In both cases we in fact get equiconsistency theorems. This strengthens results of [Rä00] and [Rä01]. is equivalent with the existence of 0^#.Mathematics Subject Classification (1991): Primary 03E55, 03E15, Secondary 03E35, 03E60